Question

four lines are drawn on a coordinate plane to form trapezoid wxyz. which statements are true about the lines? select three options. line wz has the same slope as line xy. line yx has a greater slope than line zy. line xw has a lesser slope than line yz. line zw has the same y - intercept as line yz. line xy has a lesser y - intercept than line xw.

Explanation:

Step1: Find the slope of each line

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For line \(WZ\): Let \(W(-4,1)\) and \(Z(-1,-3)\), \(m_{WZ}=\frac{-3 - 1}{-1+4}=\frac{-4}{3}\).
For line \(XY\): Let \(X(0,3)\) and \(Y(3,0)\), \(m_{XY}=\frac{0 - 3}{3-0}=- 1\).
For line \(YX\): \(m_{YX}=-1\).
For line \(ZY\): Let \(Z(-1,-3)\) and \(Y(3,0)\), \(m_{ZY}=\frac{0 + 3}{3 + 1}=\frac{3}{4}\).
For line \(XW\): Let \(X(0,3)\) and \(W(-4,1)\), \(m_{XW}=\frac{1 - 3}{-4-0}=\frac{1}{2}\).
For line \(YZ\): \(m_{YZ}=\frac{3}{4}\).

Step2: Find the \(y\) - intercept of each line

The \(y\) - intercept is the value of \(y\) when \(x = 0\).
For line \(ZW\): Using the point - slope form \(y - y_1=m(x - x_1)\) with \(W(-4,1)\) and \(m=\frac{-4}{3}\), \(y-1=\frac{-4}{3}(x + 4)\), \(y=\frac{-4}{3}x-\frac{13}{3}\), \(y\) - intercept is \(-\frac{13}{3}\).
For line \(YZ\): Using the point - slope form with \(Z(-1,-3)\) and \(m = \frac{3}{4}\), \(y+3=\frac{3}{4}(x + 1)\), \(y=\frac{3}{4}x-\frac{9}{4}\), \(y\) - intercept is \(-\frac{9}{4}\).
For line \(XY\): \(y\) - intercept is \(3\).
For line \(XW\): \(y\) - intercept is \(3\).

Step3: Check each statement

• Line \(WZ\) has slope \(\frac{-4}{3}\), line \(XY\) has slope \(-1\), so “Line \(WZ\) has the same slope as line \(XY\)” is false.
• Line \(YX\) has slope \(-1\), line \(ZY\) has slope \(\frac{3}{4}\), \(\vert-1\vert>\frac{3}{4}\), so “Line \(YX\) has a greater slope than line \(ZY\)” is true (considering the magnitude of the slope for steepness).
• Line \(XW\) has slope \(\frac{1}{2}\), line \(YZ\) has slope \(\frac{3}{4}\), \(\frac{1}{2}<\frac{3}{4}\), so “Line \(XW\) has a lesser slope than line \(YZ\)” is true.
• Line \(ZW\) has \(y\) - intercept \(-\frac{13}{3}\), line \(YZ\) has \(y\) - intercept \(-\frac{9}{4}\), so “Line \(ZW\) has the same \(y\) - intercept as line \(YZ\)” is false.
• Line \(XY\) has \(y\) - intercept \(3\), line \(XW\) has \(y\) - intercept \(3\), so “Line \(XY\) has a lesser \(y\) - intercept than line \(XW\)” is false.

Answer:

Line \(YX\) has a greater slope than line \(ZY\), Line \(XW\) has a lesser slope than line \(YZ\)